#include "boxes.h"
#include <bits/stdc++.h>
using namespace std;
#define MAXN 10000005
typedef long long ll;
ll pos[2*MAXN];
ll st[8*MAXN],lazy[8*MAXN];
ll dp_init[MAXN];
deque<pair<int,ll>> min_q;
int n,k;
ll l;
ll ans,extra,dp_s;
ll min_(){
return min_q.front().second;
}
void insert_q(pair<int,ll> p){
while(!min_q.empty()&&p.second<min_q.back().second) min_q.pop_back();
min_q.push_back(p);
}
void erase_q(int x){
if(!min_q.empty()&&min_q.front().first==x) min_q.pop_front();
}
void update(int from,int to,ll val,int v=1,int start=0,int end=2*n){
if(start==from&&end==to){
lazy[v]+=val;
st[v]+=val;
return;
}
int mid=(start+end)/2;
if(lazy[v]){
//propagate
update(start,mid,lazy[v],2*v,start,mid);
update(mid+1,end,lazy[v],2*v+1,mid+1,end);
lazy[v]=0;
}
if(to<=mid){
update(from,to,val,2*v,start,mid);
}else if(from>mid){
update(from,to,val,2*v+1,mid+1,end);
}else{
update(from,mid,val,2*v,start,mid);
update(mid+1,to,val,2*v+1,mid+1,end);
}
st[v]=min(st[2*v],st[2*v+1]);
}
ll query(int from,int to,int v=1,int start=0,int end=2*n){
if(start==from&&end==to){
return st[v];
}
int mid=(start+end)/2;
if(lazy[v]){
//propagate
update(start,mid,lazy[v],2*v,start,mid);
update(mid+1,end,lazy[v],2*v+1,mid+1,end);
lazy[v]=0;
}
if(to<=mid){
return query(from,to,2*v,start,mid);
}else if(from>mid){
return query(from,to,2*v+1,mid+1,end);
}else{
return min(query(from,mid,2*v,start,mid),query(mid+1,to,2*v+1,mid+1,end));
}
}
ll dist(ll x){
return min(x,l-x);
}
int modn(int i){
return (i<=n)?i:(i-n);
}
ll c(int i){
return dist(pos[modn(i)])+pos[i];
}
ll d(int i){
return dist(pos[modn(i+1)])-pos[i+1];
}
ll q(int s,int i){//dp[i] with start s
return c(i)+query(max(s-1,i-k),i-1);
}
ll dp(int i){
return query(i,i)-d(i);
}
long long delivery(int N, int K, int L, int positions[]) {
n=N;
k=K;
l=L;
if(N==1){
return 2*dist(positions[0]);
}
for(int i=1;i<=n;i++){
pos[i]=positions[i-1];
}
for(int i=n+1;i<=2*n;i++){
pos[i]=pos[i-n]+l;
}
dp_init[0]=0;
insert_q({0,dp_init[0]+d(0)});//base case
//update(0,0,d(0));//base case->dp[s-1]=dp[0]=0
for(int i=1;i<=2*n-1;i++){
erase_q(i-k-1);
dp_init[i]=min_()+c(i);
insert_q({i,dp_init[i]+d(i)});
//update(i,i,q(1,i)+d(i));
}
ans=dp_init[n];
//ans=dp(n);
extra=0;
dp_s=dp_init[1];
for(int s=2;s<=n;s++){
extra+=(-dp_s);
dp_s=dp_init[s]+extra;
ans=min(ans,dp_init[s+n-1]+extra);
//update(s-1,2*n-1,-dp(s-1));//0-dp[s-1]
//ans=min(ans,dp(s+n-1));
}
return ans;
}
